10.7: Divide Monomials (Part 2)
Simplify Expressions by Applying Several Properties
We'll now summarize all the properties of exponents so they are all together to refer to as we simplify expressions using several properties. Notice that they are now defined for whole number exponents.
If a, b are real numbers and m, n are whole numbers, then
| Product Property | \(a^m • a^n = a^{m + n}\) |
| Power Property | \((a^m)^n = a^{m • n}\) |
| Product to a Power Property | \((ab)^m = a^mb^m\) |
| Quotient Property | \(\dfrac{a^{m}}{a^{n}} = a^{m − n},\quad a ≠ 0,\, m > n\) |
| \(\dfrac{a^{m}}{a^{n}} = \dfrac{1}{a^{n-m}}, \quad a ≠ 0, \,n > m\) | |
| Zero Exponent Property | \(a^0 = 1, \quad a ≠ 0\) |
| Quotient to a Power Property | \(\left(\dfrac{a}{b}\right)^{m} = \dfrac{a^{m}}{b^{m}}, b ≠ 0\) |
Simplify: \(\dfrac{(x^{2})^{3}}{x^{5}}\).
Solution
| Multiply the exponents in the numerator, using the Power Property. | \(\dfrac{x^{6}}{x^{5}} \label{10.4.46}\) |
| Subtract the exponents. | \(x \label{10.4.47}\) |
Simplify: \(\dfrac{(a^{4})^{5}}{a^{9}}\).
- Answer
-
\(a^{11}\)
Simplify: \(\dfrac{(b^{5})^{6}}{b^{11}}\).
- Answer
-
\(b^{19}\)
Simplify: \(\dfrac{(m^{8})}{(m^{2})^{4}}\).
Solution
| Multiply the exponents in the numerator, using the Power Property. | \(\dfrac{m^{8}}{m^{8}} \label{10.4.48}\) |
| Subtract the exponents. | \(m^{0} \label{10.4.49}\) |
| Zero power property | 1 |
Simplify: \(\dfrac{(k^{11}}{(k^{3})^{3}}\).
- Answer
-
\(k^2\)
Simplify: \(\dfrac{(d^{23}}{(d^{4})^{6}}\).
- Answer
-
\(\frac{1}{d}\)
Simplify: \(\left(\dfrac{x^{7}}{x^{3}}\right)^{2}\).
Solution
| Remember parentheses come before exponents, and the bases are the same so we can simplify inside the parentheses. Subtract the exponents. | \((x^{7-3})^{2} \label{10.4.50}\) |
| Simplify. | \((x^{4})^{2} \label{10.4.51}\) |
| Multiply the exponents. | \(x^{8} \label{10.4.52}\) |
Simplify: \(\left(\dfrac{f^{14}}{f^{8}}\right)^{2}\).
- Answer
-
\(f^{12}\)
Simplify: \(\left(\dfrac{b^{6}}{b^{11}}\right)^{2}\).
- Answer
-
\(\frac{1}{b^{10}}\)
Simplify: \(\left(\dfrac{p^{2}}{q^{5}}\right)^{3}\).
Solution
Here we cannot simplify inside the parentheses first, since the bases are not the same.
| Raise the numerator and denominator to the third power using the Quotient to a Power Property, \(\left(\dfrac{a}{b}\right)^{m} = \dfrac{a^{m}}{b^{m}}\) | \(\dfrac{(p^{2})^{3}}{(q^{5})^{3}} \label{10.4.53}\) |
| Use the Power Property, (a m ) n = a m • n . | \(\dfrac{p^{6}}{q^{15}} \label{10.4.54}\) |
Simplify: \(\left(\dfrac{m^{3}}{n^{8}}\right)^{5}\).
- Answer
-
\(\frac{m^{15}}{n^{40}}\)
Simplify: \(\left(\dfrac{t^{10}}{u^{7}}\right)^{2}\).
- Answer
-
\(\frac{t^{20}}{u^{14}}\)
Simplify: \(\left(\dfrac{2x^{3}}{3y}\right)^{4}\).
Solution
| Raise the numerator and denominator to the fourth power using the Quotient to a Power Property. | \(\dfrac{(2x^{3})^{4}}{(3y)^{4}} \label{10.4.55}\) |
| Raise each factor to the fourth power, using the Power to a Power Property. | \(\dfrac{2^{4} (x^{3})^{4}}{3^{4} y^{4}} \label{10.4.56}\) |
| Use the Power Property and simplify. | \(\dfrac{16x^{12}}{81y^{4}} \label{10.4.57}\) |
Simplify: \(\left(\dfrac{5b}{9c^{3}}\right)^{2}\).
- Answer
-
\(\frac{25b^2}{81c^6}\)
Simplify: \(\left(\dfrac{4p^{4}}{7q^{5}}\right)^{3}\).
- Answer
-
\(\frac{64p^{12}}{343q^{15}}\)
Simplify: \(\dfrac{(y^{2})^{3} (y^{2})^{4}}{(y^{5})^{4}}\).
Solution
| Use the Power Property. | \(\dfrac{(y^{6})(y^{8})}{y^{20}} \label{10.4.58}\) |
| Add the exponents in the numerator, using the Product Property. | \(\dfrac{y^{14}}{y^{20}} \label{10.4.59}\) |
| Use the Quotient Property. | \(\dfrac{1}{y^{6}} \label{10.4.60}\) |
Simplify: \(\dfrac{(y^{4})^{4} (y^{3})^{5}}{(y^{7})^{6}}\).
- Answer
-
\(\frac{1}{y^{11}}\)
Simplify: \(\dfrac{(3x^{4})^{2} (x^{3})^{4}}{(x^{5})^{3}}\).
- Answer
-
\(9x^5\)
Divide Monomials
We have now seen all the properties of exponents. We'll use them to divide monomials. Later, you'll use them to divide polynomials.
Find the quotient: 56x 5 ÷ 7x 2 .
Solution
| Rewrite as a fraction. | \(\dfrac{56x^{5}}{7x^{2}} \label{10.4.61}\) |
| Use fraction multiplication to separate the number part from the variable part. | \(\dfrac{56}{7} \cdot \dfrac{x^{5}}{x^{2}} \label{10.4.62}\) |
| Use the Quotient Property. | \(8x^{3} \label{10.4.63}\) |
Find the quotient: 63x 8 ÷ 9x 4 .
- Answer
-
\(7x^4\)
Find the quotient: 96y 11 ÷ 6y 8 .
- Answer
-
\(16y^3\)
When we divide monomials with more than one variable, we write one fraction for each variable.
Find the quotient: \(\dfrac{42x^{2} y^{3}}{−7xy^{5}}\).
Solution
| Use fraction multiplication. | \(\dfrac{42}{-7} \cdot \dfrac{x^{2}}{x} \cdot \dfrac{y^{3}}{y^{5}} \label{10.4.64}\) |
| Simplify and use the Quotient Property. | \(-6 \cdot x \cdot \dfrac{1}{y^{2}} \label{10.4.65}\) |
| Multiply. | \(- \dfrac{6x}{y^{2}} \label{10.4.66}\) |
Find the quotient: \(\dfrac{-84x^{8} y^{3}}{7x^{10} y^{2}}\).
- Answer
-
\(-\frac{12y}{x^2}\)
Find the quotient: \(\dfrac{-72a^{4} b^{5}}{−8a^{9} b^{5}}\).
- Answer
-
\(\frac{9}{a^5}\)
Find the quotient: \(\dfrac{24a^{5} b^{3}}{48ab^{4}}\).
Solution
| Use fraction multiplication. | \(\dfrac{24}{48} \cdot \dfrac{a^{5}}{a} \cdot \dfrac{b^{3}}{b^{4}} \label{10.4.67}\) |
| Simplify and use the Quotient Property. | \(\dfrac{1}{2} \cdot a^{4} \cdot \dfrac{1}{b} \label{10.4.68}\) |
| Multiply. | \(\dfrac{a^{4}}{2b} \label{10.4.69}\) |
Find the quotient: \(\dfrac{16a^{7} b^{6}}{24ab^{8}}\).
- Answer
-
\(\frac{2a^6}{3b^2}\)
Find the quotient: \(\dfrac{27p^{4} q^{7}}{-45p^{12} q}\).
- Answer
-
\(-\frac{3q^6}{5p^8}\)
Once you become familiar with the process and have practiced it step by step several times, you may be able to simplify a fraction in one step.
Find the quotient: \(\dfrac{14x^{7} y^{12}}{21x^{11} y^{6}}\).
Solution
| Simplify and use the Quotient Property. | \(\dfrac{2y^{6}}{3x^{4}} \label{10.4.70}\) |
Be very careful to simplify \(\dfrac{14}{21}\) by dividing out a common factor, and to simplify the variables by subtracting their exponents.
Find the quotient: \(\dfrac{28x^{5} y^{14}}{49x^{9} y^{12}}\).
- Answer
-
\(\frac{4y^2}{7x^4}\)
Find the quotient: \(\dfrac{30m^{5} n^{11}}{48m^{10} n^{14}}\).
- Answer
-
\(\frac{5}{8m^5 n^3}\)
In all examples so far, there was no work to do in the numerator or denominator before simplifying the fraction. In the next example, we'll first find the product of two monomials in the numerator before we simplify the fraction.
Find the quotient: \(\dfrac{(3x^{3} y^{2})(10x^{2} y^{3})}{6x^{4} y^{5}}\).
Solution
Remember, the fraction bar is a grouping symbol. We will simplify the numerator first.
| Simplify the numerator. | \(\dfrac{30x^{5} y^{5}}{6x^{4} y^{5}} \label{10.4.71}\) |
| Simplify, using the Quotient Rule. | \(5x \label{10.4.72}\) |
Find the quotient: \(\dfrac{(3x^{4} y^{5})(8x^{2} y^{5})}{12x^{5} y^{8}}\).
- Answer
-
\(2xy^2\)
Find the quotient: \(\dfrac{(-6a^{6} b^{9})(-8a^{5} b^{8})}{-12a^{10} b^{12}}\).
- Answer
-
\(-4ab^5\)
Simplify a Quotient
Zero Exponent
Quotient Rule
Polynomial Division
Polynomial Division 2
Practice Makes Perfect
Simplify Expressions Using the Quotient Property of Exponents
In the following exercises, simplify.
- \(\dfrac{4^{8}}{4^{2}}\)
- \(\dfrac{3^{12}}{3^{4}}\)
- \(\dfrac{x^{12}}{x^{3}}\)
- \(\dfrac{u^{9}}{u^{3}}\)
- \(\dfrac{r^{5}}{r}\)
- \(\dfrac{y^{4}}{y}\)
- \(\dfrac{y^{4}}{y^{20}}\)
- \(\dfrac{x^{10}}{x^{30}}\)
- \(\dfrac{10^{3}}{10^{15}}\)
- \(\dfrac{r^{2}}{r^{8}}\)
- \(\dfrac{a}{a^{9}}\)
- \(\dfrac{2}{2^{5}}\)
Simplify Expressions with Zero Exponents
In the following exercises, simplify.
- 5 0
- 10 0
- a 0
- x 0
- −7 0
- −4 0
- (a) (10p) 0 (b) 10p 0
- (a) (3a) 0 (b) 3a 0
- (a) (−27x 5 y) 0 (b) −27x 5 y 0
- (a) (−92y 8 z) 0 (b) −92y 8 z 0
- (a) 15 0 (b) 15 1
- (a) −6 0 (b) −6 1
- 2 • x 0 + 5 • y 0
- 8 • m 0 − 4 • n 0
Simplify Expressions Using the Quotient to a Power Property
In the following exercises, simplify.
- \(\left(\dfrac{3}{2}\right)^{5}\)
- \(\left(\dfrac{4}{5}\right)^{3}\)
- \(\left(\dfrac{m}{6}\right)^{3}\)
- \(\left(\dfrac{p}{2}\right)^{5}\)
- \(\left(\dfrac{x}{y}\right)^{10}\)
- \(\left(\dfrac{a}{b}\right)^{8}\)
- \(\left(\dfrac{a}{3b}\right)^{2}\)
- \(\left(\dfrac{2x}{y}\right)^{4}\)
Simplify Expressions by Applying Several Properties
In the following exercises, simplify.
- \(\dfrac{(x^{2})^{4}}{x^{5}}\)
- \(\dfrac{(y^{4})^{3}}{y^{7}}\)
- \(\dfrac{(u^{3})^{4}}{u^{10}}\)
- \(\dfrac{(y^{2})^{5}}{y^{6}}\)
- \(\dfrac{y^{8}}{(y^{5})^{2}}\)
- \(\dfrac{p^{11}}{(p^{5})^{3}}\)
- \(\dfrac{r^{5}}{(r^{4} \cdot r}\)
- \(\dfrac{a^{3} \cdot a^{4}}{(a^{7}}\)
- \(\left(\dfrac{x^{2}}{x^{8}}\right)^{3}\)
- \(\left(\dfrac{u}{u^{10}}\right)^{2}\)
- \(\left(\dfrac{a^{4} \cdot a^{6}}{a^{3}}\right)^{2}\)
- \(\left(\dfrac{x^{3 \cdot x^{8}}}{x^{4}}\right)^{3}\)
- \(\dfrac{(y^{3})^{5}}{(y^{4})^{3}}\)
- \(\dfrac{(z^{6})^{2}}{(z^{2})^{4}}\)
- \(\dfrac{(x^{3})^{6}}{(x^{4})^{7}}\)
- \(\dfrac{(x^{4})^{8}}{(x^{5})^{7}}\)
- \(\left(\dfrac{2r^{3}}{5s}\right)^{4}\)
- \(\left(\dfrac{3m^{2}}{4n}\right)^{3}\)
- \(\left(\dfrac{3y^{2} \cdot y^{5}}{y^{15} \cdot y^{8}}\right)^{0}\)
- \(\left(\dfrac{15z^{4} \cdot z^{9}}{0.3z^{2}}\right)^{0}\)
- \(\dfrac{(r^{2})^{5} (r^{4})^{2}}{(r^{3})^{7}}\)
- \(\dfrac{(p^{4})^{2} (p^{3})^{5}}{(p^{2})^{9}}\)
- \(\dfrac{(3x^{4})^{3} (2x^{3})^{2}}{(6x^{5})^{2}}\)
- \(\dfrac{(-2y^{3})^{4} (3y^{4})^{2}}{(-6y^{3})^{2}}\)
Divide Monomials
In the following exercises, divide the monomials.
- 48b 8 ÷ 6b 2
- 42a 14 ÷ 6a 2
- 36x 3 ÷ (−2x 9 )
- 20u 8 ÷ (−4u 6 )
- \(\dfrac{18x^{3}}{9x^{2}}\)
- \(\dfrac{36y^{9}}{4y^{7}}\)
- \(\dfrac{-35x^{7}}{-42x^{13}}\)
- \(\dfrac{18x^{5}}{-27x^{9}}\)
- \(\dfrac{18r^{5} s}{3r^{3} s^{9}}\)
- \(\dfrac{24p^{7} q}{6p^{2} q^{5}}\)
- \(\dfrac{8mn^{10}}{64mn^{4}}\)
- \(\dfrac{10a^{4} b}{50a^{2} b^{6}}\)
- \(\dfrac{-12x^{4} y^{9}}{15x^{6} y^{3}}\)
- \(\dfrac{48x^{11} y^{9} z^{3}}{36x^{6} y^{8} z^{5}}\)
- \(\dfrac{64x^{5} y^{9} z^{7}}{48x^{7} y^{12} z^{6}}\)
- \(\dfrac{(10u^{2} v)(4u^{3} v^{6})}{5u^{9} v^{2}}\)
- \(\dfrac{(6m^{2} n)(5m^{4} n^{3})}{3m^{10} n^{2}}\)
- \(\dfrac{(6a^{4} b^{3})(4ab^{5})}{(12a^{8} b)(a^{3} b)}\)
- \(\dfrac{(4u^{5} v^{4})(15u^{8} v)}{(12u^{3} v)(u^{6} v)}\)
Mixed Practice
- (a) 24a 5 + 2a 5 (b) 24a 5 − 2a 5 (c) 24a 5 • 2a 5 (d) 24a 5 ÷ 2a 5
- (a) 15n 10 + 3n 10 (b) 15n 10 − 3n 10 (c) 15n 10 • 3n 10 (d) 15n 10 ÷ 3n 10
- (a) p 4 • p 6 (b) (p 4 ) 6
- (a) q 5 • q 3 (b) (q 5 ) 3
- (a) \(\dfrac{ y^{3}}{y}\) (b) \(\dfrac{y}{y^{3}}\)
- (a) \(\dfrac{z^{6}}{z^{5}}\) (b) \(\dfrac{z^{5}}{z^{6}}\)
- (8x 5 )(9x) ÷ 6x 3
- (4y 5 )(12y 7 ) ÷ 8y 2
- \(\dfrac{27a^{7}}{3a^{3}} + \dfrac{54a^{9}}{9a^{5}}\)
- \(\dfrac{32c^{11}}{4c^{5}} + \dfrac{42c^{9}}{6c^{3}}\)
- \(\dfrac{32y^{5}}{8y^{2}} - \dfrac{60y^{10}}{5y^{7}}\)
- \(\dfrac{48x^{6}}{6x^{4}} - \dfrac{35x^{9}}{7x^{7}}\)
- \(\dfrac{63r^{6} s^{3}}{9r^{4} s^{2}} - \dfrac{72r^{2} s^{2}}{6s}\)
- \(\dfrac{56y^{4} z^{5}}{7y^{3} z^{3}} - \dfrac{45y^{2} z^{2}}{5y}\)
Everyday Math
- Memory One megabyte is approximately 10 6 bytes. One gigabyte is approximately 10 9 bytes. How many megabytes are in one gigabyte?
- Memory One megabyte is approximately 10 6 bytes. One terabyte is approximately 10 12 bytes. How many megabytes are in one terabyte?
Writing Exercises
- Vic thinks the quotient \(\dfrac{x^{20}}{x^{4}}\) simplifies to x 5 . What is wrong with his reasoning?
- Mai simplifies the quotient \(\dfrac{y^{3}}{y}\) by writing \(\dfrac{y^{3}}{y}\) = 3. What is wrong with her reasoning?
- When Dimple simplified −3 0 and (−3) 0 she got the same answer. Explain how using the Order of Operations correctly gives different answers.
- Roxie thinks n 0 simplifies to 0. What would you say to convince Roxie she is wrong?
Self Check
(a) After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
(b) On a scale of 1–10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?
Contributors and Attributions
-
Lynn Marecek (Santa Ana College) and MaryAnne Anthony-Smith (Formerly of Santa Ana College). This content is licensed under Creative Commons Attribution License v4.0 "Download for free at http://cnx.org/contents/fd53eae1-fa2...49835c3c@5.191 ."